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 Probabilistic and deterministic algorithms for space multidimensional irregular porous media equation.
 Nadia Belaribi 1, 2, François Cuvelier 2
 For the EPFL Lausanne, SFB 701 Universitaet Bielefeld collaboration(s)
 (2012-08-20)
 The object of this paper is a multi-dimensional generalized porous media equation (PDE) with not smooth and possibly discontinuous coefficient $\beta$, which is well-posed as an evolution problem in $L^1(\mathbb{R}^d)$. This work continues the study related to the one-dimensional case by the same authors. One expects that a solution of the mentioned PDE can be represented through the solution (in law) of a non-linear stochastic differential equation (NLSDE). A classical tool for doing this is a uniqueness argument for some Fokker-Planck type equations with measurable coefficients. When $\beta$ is possibly discontinuous, this is often possible in dimension $d = 1$. If $d > 1$, this problem is more complex than for $d = 1$. However, it is possible to exhibit natural candidates for the probabilistic representation and to use them for approximating the solution of (PDE) through a stochastic particle algorithm. We compare it with some numerical deterministic techniques that we have implemented adapting the method of a paper of Cavalli et al. whose convergence was established when $\beta$ is Lipschitz. Special emphasis is also devoted to the case when the initial condition is radially symmetric. On the other hand assuming that $\beta$ is continuous (even though not smooth), one provides existence results for a mollified version of the (NLSDE) and a related partial integro-differential equation, even if the initial condition is a general probability measure.
 1: Unité de Mathématiques Appliquées (UMA) ENSTA ParisTech 2: Laboratoire Analyse, Géométrie et Application (LAGA) CNRS : UMR7539 – Université Paris XIII - Paris Nord – Université Paris VIII - Vincennes Saint-Denis
 Domain : Mathematics/Probability
 Keywords: Stochastic particle algorithm – porous media equation – monotonicity – stochastic differential equations – non-parametric density estimation – kernel estimator
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 hal-00723821, version 1 http://hal.inria.fr/hal-00723821 oai:hal.inria.fr:hal-00723821 From: Francesco Russo <> Submitted on: Monday, 20 August 2012 10:44:26 Updated on: Monday, 20 August 2012 11:06:05